Some of the applications of the Bures metric include that given a target error, it allows the calculation of the minimum number of measurements to distinguish two different states [1] and the use of the volume element as a candidate for the Jeffreys prior probability density [2] for mixed quantum states.

The actual computation of the Bures metric is not evident from the definition, so, some formulas were developed for that purpose. Dittmann obtained the following formulas for the quadratic form of the Bures metric, valid for 2x2 and 3x3 systems, respectively

Another important formula is the one found by Hübner. This formula is written in terms of the eigenvectors and eigenvalues of the density matrix and reads